Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs

Abstract

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique in $O(m + n^{4.5(1-\alpha)})$ expected time if the geometric representation of the hyperbolic random graph is given or in $O(m + n^{6(1-\alpha)})$ expected time if the geometric representation is not given. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently.